Is it normal that i stay with my boyfriend if he does this?

My boyfriend and I have been together since November of 2010. I met him in college and he's 2 grades above me. We weren't officially boyfriend girlfriend until January. But in between we've had some serious problems. Me and him have discussed how we're monogamous since the beginning. But right before Xmas break, he tells me that he's in love with some other girl that he met once at a party here at college, but she's from Alabama and was visiting. So he's only met her once, but they text all the time. This girl I knew about, but he had said she was just a friend and I thought nothing about it. So I break it off, upset. After Xmas break, he comes crawling back to me and after some thought, I let him back into my life. Then we officially get together in January and we're boyfriend girlfriend. But I still see him texting this girl he "supposedly got over" and he says they're just friends now. But I know thats bull. So I ask him to stop texting her, especially in front of me. And he finally comes out again a month later that he loves her and doesn't know what to do. I finally make him realize this is a fake fantasy relationship. Apparently he met her on the internet and has never even met the girl. So with much convincing on his part, I decide to stay with him but he has to break it off completely and delete this girl from his life. So all is well, and I feel like he did that cause things got better. But then he comes out again a few months later that he continued to text her, and said that he loved her but didn't mean it cause he didn't know what to say. He officially blocked her number and said he is so disgusted and mad at her and never wants to talk to her again due to everything and says all he wants to do is be with me. From the minute he realized he love me a month ago, he officially stopped talking to her and would never ever hurt me again. Do I stay with him and work on our trust issues, cause he seems so willing to be with me, or do I dump him because he has done this so many times? I feel stupid for giving him that many chances, but obviously there is a part of him I really love and respect. He is the perfect boyfriend in all other instances for the most part. But I don't know what to do. Please help.